On minimal non-abelian subgroups in finite p-groups

We determine the non-abelian finite p-groups G with C(G)(x) <= H for any minimal non-abelian subgroup H of G and each x is an element of H - Z(G) (Theorem 1.1). This solves Problem 757 of Berkovich [1]. We also classify the non-abelian finite p-groups G such that whenever A is a maximal subgroup of any minimal non-abelian subgroup H in G, then A is also a maximal abelian subgroup in G (Theorem 1.2), and this solves another problem of Berkovich [1]. Finally, we generalize a result of Blackburn [4] concerning finite p-groups in which the non-normal subgroups have non-trivial intersection (Theorem 1.3 and Corollary 1.4).